Question

A right circular cylinder and a right circular cone have equal bases and equal heights. If their curved surfaces are in the ratio $8:5$, determine the ratio of the radius of the base to the height .

Answer 1

Given,

A right circular cylinder and a right circular cone have equal bases and equal heights, and their curved surface area is in the ratio $8:5$

Let, the radius of both cylinder and cone be $r$

And, the hight of both cylinder and cone be $h$

Now,

Curved surface area of cylinder $S_1=2\pi rh$

Curved surface area of cone $S_2=\pi r\sqrt{r^2+h^2}$

And, given that

$\frac{S_1}{S_2}=\frac{8}{5}$

$\frac{2\pi rh}{\pi r\sqrt{r^2+h^2}}=\frac{8}{5}$

$\frac{2h}{\sqrt{r^2+h^2}}=\frac{8}{5}$

$10h=8\sqrt{r^2+h^2}$

squaring both sides.

$(10h{)}^2=(8{)}^2(r^2+h^2)$

$(100h^2=64(r^2+h^2)$

$100h^2=64r^2+64h^2$

$100h^2-64h^2=64r^2$

$36h^2=64r^2$

$\frac{r^2}{h^2}=\frac{36}{64}$

$\frac{r}{h}=\frac{6}{8}=\frac{3}{4}$

Hence, the ratio of radius of the base and hight is

$r:h=3:4$